Laplacians on Self-Similar Fractals and Their Spectral Zeta Functions: Complex Dynamics and Renormalization

Transcription

1 Laplacians on Self-Similar Fractals and Their Spectral Zeta Functions: Complex Dynamics and Renormalization Michel L. Lapidus University of California, Riverside Joint work with Nishu Lal University of California, Riverside and Scripps College, Claremont October 27, 2012 Michel L. Lapidus University of California, Riverside Laplacians on Self-Similar Joint work Fractals withand Nishu Their LalSpectral University Zeta of Functions: California, Riverside Complex October Dynamics and Scripps 27, 2012 and College, Renormalizatio 1 Claremo / 41

2 The key points Spectral zeta functions of Laplacians on fractals Factorization of the spectral zeta functions, usually in terms of well known zeta functions Michel L. Lapidus University of California, Riverside Laplacians on Self-Similar Joint work Fractals withand Nishu Their LalSpectral University Zeta of Functions: California, Riverside Complex October Dynamics and Scripps 27, 2012 and College, Renormalizatio 2 Claremo / 41

3 The spectral zeta function Definition The spectral zeta function of a positive self-adjoint operator L with compact resolvent (and hence, with discrete spectrum) is given for Re(s) large enough by ζ L (s) = (κ j ) s/2, j=1 where the positive real numbers κ j are the eigenvalues of the operator written in nonincreasing order and counted according to their multiplicities. ichel L. Lapidus University of California, Riverside lapidus@math.ucr.edu Laplacians on Self-Similar Joint work Fractals withand Nishu Their LalSpectral University Zeta of Functions: California, Riverside Complex October Dynamics and Scripps 27, 2012 and College, Renormalizatio 3 Claremo / 41

4 The Laplacian on the unit interval Consider the Dirichlet Laplacian A = d 2 dx 2 on [0,1]. Eigenvalues: λ = π 2 j 2, j = 1,2,... The spectral zeta function is ζ A (s) = j=1 (π2 j 2 ) s/2 = π s j=1 j s = π s ζ(s). Michel L. Lapidus University of California, Riverside lapidus@math.ucr.edu Laplacians on Self-Similar Joint work Fractals withand Nishu Their LalSpectral University Zeta of Functions: California, Riverside Complex October Dynamics and Scripps 27, 2012 and College, Renormalizatio 4 Claremo / 41

5 Fractal string Definition A fractal string L is a countable collection of disjoint open intervals of length l k. Definition The geometric zeta function of the fractal string is given by ζ L (s) = (l k ) s. k=1 ichel L. Lapidus University of California, Riverside lapidus@math.ucr.edu Laplacians on Self-Similar Joint work Fractals withand Nishu Their LalSpectral University Zeta of Functions: California, Riverside Complex October Dynamics and Scripps 27, 2012 and College, Renormalizatio 5 Claremo / 41

6 Associated spectal zeta function Consider L = k=1 I k with I k = l k and A = d 2 dx 2 be the Dirichlet Laplacian on L. σ(a) = k=1 σ(a; I k) = { π2 n 2 : n 1, k 1}. l 2 k The spectral zeta function of A is ζ A (s) = k,n 1 ( π2 n 2 ) s/2 = π s l 2 k=1 ls k n=1 n s. k Theorem (Lap, 1992) ζ A (s) = π s ζ L (s)ζ(s), where ζ(s) is the Riemann zeta function. Remark: See, e.g., [Lap, 1992], [Lap Mai, 1995], [Lap Pom, 1993] and [Lap vfran, 2006] for various applications of this formula. ichel L. Lapidus University of California, Riverside lapidus@math.ucr.edu Laplacians on Self-Similar Joint work Fractals withand Nishu Their LalSpectral University Zeta of Functions: California, Riverside Complex October Dynamics and Scripps 27, 2012 and College, Renormalizatio 6 Claremo / 41

7 The Sierpinski Gasket The Sierpinski gasket SG, is a classical example of a self-similar fractal on which the Laplacian is widely explored. (See, e.g., [Kig, 2001] or [Str, 2006].) Michel L. Lapidus University of California, Riverside lapidus@math.ucr.edu Laplacians on Self-Similar Joint work Fractals withand Nishu Their LalSpectral University Zeta of Functions: California, Riverside Complex October Dynamics and Scripps 27, 2012 and College, Renormalizatio 7 Claremo / 41

8 Michel L. Lapidus University of California, Riverside Laplacians on Self-Similar Joint work Fractals withand Nishu Their LalSpectral University Zeta of Functions: California, Riverside Complex October Dynamics and Scripps 27, 2012 and College, Renormalizatio 8 Claremo / 41

9 The normalized discrete Laplacian on SG The normalized discrete Laplacian on Γ m is defined by for u : V m R with u vanishing on V 0. m u(x) := 1 (u(x) u(y)) 4 x m y Then the Laplacian on the Sierpinski gasket is defined as a limit u(x) = lim m 5m m u(x). Michel L. Lapidus University of California, Riverside lapidus@math.ucr.edu Laplacians on Self-Similar Joint work Fractals withand Nishu Their LalSpectral University Zeta of Functions: California, Riverside Complex October Dynamics and Scripps 27, 2012 and College, Renormalizatio 9 Claremo / 41

10 Decimation method In analyzing the eigenvalue spectrum of the Laplacian on SG, we first consider the eigenvalue equation associated with each m : m u = λu. The decimation method is a process that relates the eigenvalues of successive graph Laplacians by the polynomial R(z) = z(5 4z), in particular the inverse image: R (z) = z 8 R + (z) = z 8. (See, e.g., [Ram, 1984], [Ram Tou, 1983], [Shi, 1996], [Fuk Shi, 1992], or [Str, 2006].) Michel L. Lapidus University of California, Riverside lapidus@math.ucr.edu Laplacians on Self-Similar Joint work Fractals withand Nishu Their LalSpectral University Zeta of Functions: California, Riverside Complex October Dynamics and Scripps 27, 2012 and College, Renormalizatio 10 Claremo / 41

11 Eigenvalue diagram Operators Eigenvalues 0 0 [1] 3 2 [2] 1 0 [1] [2] [3] [1] 5 4 R + (3/4) R (3/4) [2] [2] 3 4 [3] [6] 5 [1] [1] 5 4 R + R(3/4) + R + R(3/4) [2] [2] R(3/4) + R(3/4) [3] [3] 3 4 [6] 1 2 R(5/4) + R(5/4) [1] [1] 3 2 [15] 5 4 [4] 4 0 [1] 5 4 R(3/4) R(3/4) + [6] [6] 3 4 [15] 1 2 R(5/4) R(5/4) + [4] [4] Michel L. Lapidus University of California, Riverside lapidus@math.ucr.edu Laplacians on Self-Similar Joint work Fractals withand Nishu Their LalSpectral University Zeta of Functions: California, Riverside Complex October Dynamics and Scripps 27, 2012 and College, Renormalizatio 11 Claremo / 41

12 Factorization of the spectral zeta function Definition (Tep, 2007) Let R be a polynomial of degree N satisfying R(0) = 0, c := R (0) > 1, and with Julia set J [0, ). Then the zeta function of R of degree N is defined by ζ R,z0 (s) = lim (c n z) s 2, for Re(s) > d R := 2 log N log c. Theorem (Tep, 2007) n z R n {z 0} The spectral zeta function of the Laplacian on SG is s ( ) s ( ) 3 1 ζ µ (s) = ζ R, 3 (s) s ζ s R, s s 2 where R(z) = z(5 4z). Furthermore, there exists ɛ > 0 such that ζ µ (s) has a { meromorphic continuation } for Re(s) > ɛ with poles contained in 2inπ log 9+2inπ, : n Z. log 5 log 5 ichel L. Lapidus University of California, Riverside lapidus@math.ucr.edu Laplacians on Self-Similar Joint work Fractals withand Nishu Their LalSpectral University Zeta of Functions: California, Riverside Complex October Dynamics and Scripps 27, 2012 and College, Renormalizatio 12 Claremo / 41

13 Now we discuss the factorization of the spectral zeta function of the fractal Sturm Liouville operator on the half-line (a model introduced by C. Sabot in [Sab, 1998, 2001, 2003, & 2005]). We will extend the results on the previous slides to several complex variables. Michel L. Lapidus University of California, Riverside Laplacians on Self-Similar Joint work Fractals withand Nishu Their LalSpectral University Zeta of Functions: California, Riverside Complex October Dynamics and Scripps 27, 2012 and College, Renormalizatio 13 Claremo / 41

14 R + as a fractal Consider the two contraction mappings Ψ 1 and Ψ 2 on I = [0, 1] defined for a given α (0, 1) by Ψ 1 (x) = αx Ψ 2 (x) = 1 (1 α)(1 x). Blow-up of the interval I : I <n> = Ψ n 1 (I ) = [0, α n ]. I <n> = i 1...i n Ψ i1...i n (I <n> ) where (i 1,..., i n ) {1, 2} n. Michel L. Lapidus University of California, Riverside lapidus@math.ucr.edu Laplacians on Self-Similar Joint work Fractals withand Nishu Their LalSpectral University Zeta of Functions: California, Riverside Complex October Dynamics and Scripps 27, 2012 and College, Renormalizatio 14 Claremo / 41

15 On I The self-similar measure on I : m is a measure such that for all f C([0, 1]), 1 Define H <0> = d the domain 0 fdm = b dm 1 0 f Ψ 1 dm + (1 b) 1 0 f Ψ 2 dm. d dx with Dirichlet boundary condition by H <0>f = g on {f L 2 (I, m), g L 2 (I, m), f (x) = cx +d + x 0 y g(z)dm(z)dy, f (0) = f (1) = 0}. 0 The operator H <0> is the infinitesimal generator associated with the Dirichlet form (a, D) given by where a(f, g) = 1 0 f g dx, f, g D, D = {f L 2 (I, m) : f L 2 (I, dx)}. Michel L. Lapidus University of California, Riverside lapidus@math.ucr.edu Laplacians on Self-Similar Joint work Fractals withand Nishu Their LalSpectral University Zeta of Functions: California, Riverside Complex October Dynamics and Scripps 27, 2012 and College, Renormalizatio 15 Claremo / 41

16 On the extension I <n> I < > = R + Recall: I <n> = i 1...i n Ψ i1...i n (I <n> ) where (i 1,..., i n ) {1, 2} n. We extend the definitions of the measure and the Dirichlet forms to I <n>. m <n> is defined by I <n> fdm <n> = (1 α) n I f Ψ n 1 dm, f C(I <n>). (a <n>, D <n> ) is defined by where a <n> (f, f ) = α n 0 (f ) 2 dx = α n a(f Ψ n 1 ), f D <n>, D <n> = { f L 2 (I <n>, m <n> ) : f exists and f L 2 (I <n>, dx) }. Michel L. Lapidus University of California, Riverside lapidus@math.ucr.edu Laplacians on Self-Similar Joint work Fractals withand Nishu Their LalSpectral University Zeta of Functions: California, Riverside Complex October Dynamics and Scripps 27, 2012 and College, Renormalizatio 16 Claremo / 41

17 The second order differential operator H <n> H < > We define: D = {f L 2 (I <n>, m <n>), g L 2 (I <n>, m <n>), f (x) = ux + v + x y g(z)dm<n>(z)dy, f (0) = f 0 0 (α n ) = 0}. Definition For any f D, we define the operator H <n> = conditions on I <n> by H <m>f = g. d d dm <n> dx with Dirichlet boundary The Sturm Liouville operator H < > on [0, ) is viewed as a limit of the sequence of operators H <n> = d d with Dirichlet boundary condition on dx I<n> = [0, α n ]. dm <n> ichel L. Lapidus University of California, Riverside lapidus@math.ucr.edu Laplacians on Self-Similar Joint work Fractals withand Nishu Their LalSpectral University Zeta of Functions: California, Riverside Complex October Dynamics and Scripps 27, 2012 and College, Renormalizatio 17 Claremo / 41

18 Eigenvalue Problem Consider the eigenvalue problem H <n> f = d d dm <n> dx f = λf of the Sturm Liouville operator with Dirichlet boundary condition on I <n>. Michel L. Lapidus University of California, Riverside lapidus@math.ucr.edu Laplacians on Self-Similar Joint work Fractals withand Nishu Their LalSpectral University Zeta of Functions: California, Riverside Complex October Dynamics and Scripps 27, 2012 and College, Renormalizatio 18 Claremo / 41

19 Renormalization map The study of the eigenvalue problem revolves around a map, called the renormalization map, which is initially defined on a space of quadratic forms associated with the fractal. The propagator of the differential equation is very useful in producing this rational map, ρ([x, y, z]) = [x(x + δ 1 y) δ 1 z 2, δy(x + δ 1 y) δz 2, z 2 ] defined on P 2 (C). (See [Sab, 2001].) Here, [x, y, z] denote the homogeneous coordinate of a point in P 2 (C), where (x, y, z) C 3 is identified with (βx, βy, βz) for any β C, β 0. ichel L. Lapidus University of California, Riverside lapidus@math.ucr.edu Laplacians on Self-Similar Joint work Fractals withand Nishu Their LalSpectral University Zeta of Functions: California, Riverside Complex October Dynamics and Scripps 27, 2012 and College, Renormalizatio 19 Claremo / 41

20 The invariant curve We introduce the invariant curve φ(x) which is holomorphic on C and such that for all λ C, ρ(φ(λ)) = φ(γλ), where γ := 1 α(1 α). (Note that γ 4; in particular, γ > 1.) Michel L. Lapidus University of California, Riverside lapidus@math.ucr.edu Laplacians on Self-Similar Joint work Fractals withand Nishu Their LalSpectral University Zeta of Functions: California, Riverside Complex October Dynamics and Scripps 27, 2012 and College, Renormalizatio 20 Claremo / 41

21 The spectrum of H <n> on I <n> The dynamics of the renormalization map ρ plays a key role in calculating the spectrum of the operators H <n>. Let D = {[x, y, z] : x + δ 1 y = 0}. The set D is part of the Fatou set of ρ. Let S be the time intersections of the curve φ(γ 1 λ) with D, S = {λ C : φ(γ 1 λ) D}. Michel L. Lapidus University of California, Riverside lapidus@math.ucr.edu Laplacians on Self-Similar Joint work Fractals withand Nishu Their LalSpectral University Zeta of Functions: California, Riverside Complex October Dynamics and Scripps 27, 2012 and College, Renormalizatio 21 Claremo / 41

22 The spectrum of H <0> Theorem (Sab, 2005) Let S p = γ p S with p Z. Then the spectrum of H <0> on I = I <0> is p=0 S p and the spectrum of H < > on R + is p= S p. Moreover, for any n 0, the spectrum of H <n> is equal to p= n S p. The set S is non-empty and contained in R +. We arrange the elements of S as 0 λ 1 λ 2... λ k... ichel L. Lapidus University of California, Riverside lapidus@math.ucr.edu Laplacians on Self-Similar Joint work Fractals withand Nishu Their LalSpectral University Zeta of Functions: California, Riverside Complex October Dynamics and Scripps 27, 2012 and College, Renormalizatio 22 Claremo / 41

23 The spectrum of H < > The eigenvalue diagram associated with each λ j S has the following form:.... γ 2 λ 1 γ 2 λ 2 γ 2 λ 3 γ 2 λ 4 γ 1 λ 1 γ 1 λ 2 γ 1 λ 3 γ 1 λ 4 λ 1 λ 2 λ 3 λ 4 γλ 1 γλ 2 γλ 3 γλ 4 γ 2 λ 1 γ 2 λ 2 γ 2 λ 3 γ 2 λ Michel L. Lapidus University of California, Riverside lapidus@math.ucr.edu Laplacians on Self-Similar Joint work Fractals withand Nishu Their LalSpectral University Zeta of Functions: California, Riverside Complex October Dynamics and Scripps 27, 2012 and College, Renormalizatio 23 Claremo / 41

24 The zeta function associated to the renormalization map We introduce a multivariable analog of the polynomial zeta function. Definition (Lal Lap, 2012) We define the zeta function of the renormalization map ρ to be ζ ρ (s) = (γ p λ) s 2, (1) for Re(s) sufficiently large. p=0 {λ C: ρ p (φ(γ (p+1) λ)) D} ichel L. Lapidus University of California, Riverside lapidus@math.ucr.edu Laplacians on Self-Similar Joint work Fractals withand Nishu Their LalSpectral University Zeta of Functions: California, Riverside Complex October Dynamics and Scripps 27, 2012 and College, Renormalizatio 24 Claremo / 41

25 The spectral zeta function of H <0> Recall that given an integer n 0, the spectral zeta function ζ H<n> (s) of H <n> on [0, α n ] is ζ H<n> (s) = (γ p λ) s 2. λ S p= n Theorem (Lal Lap, 2012) The zeta function ζ ρ (s) of the renormalization map ρ is equal to the spectral zeta function ζ H<0> (s) = λ S p=0 (γp λ) s 2 of H <0> (s): ζ ρ (s) = ζ H<0> (s). ichel L. Lapidus University of California, Riverside lapidus@math.ucr.edu Laplacians on Self-Similar Joint work Fractals withand Nishu Their LalSpectral University Zeta of Functions: California, Riverside Complex October Dynamics and Scripps 27, 2012 and College, Renormalizatio 25 Claremo / 41

26 The factorization formulas for ζ <n> Proposition (Lal Lap, 2012) For n 0 and Re(s) sufficiently large and positive, we have ζ H<n> (s) = (γn ) s 2 1 γ s 2 (λ j ) s 2. (2) Hence, ζ H<n> (s) = (γn ) s 2 1 γ s 2 ζ S(s), where ζ S (s) := j=1 (λ j) s 2 (for Re(s) large enough) or is given by its meromorphic continuation thereof. In the sequel, ζ S (s) is called the geometric zeta function of the generating set S. j=1 ichel L. Lapidus University of California, Riverside lapidus@math.ucr.edu Laplacians on Self-Similar Joint work Fractals withand Nishu Their LalSpectral University Zeta of Functions: California, Riverside Complex October Dynamics and Scripps 27, 2012 and College, Renormalizatio 26 Claremo / 41

27 The spectral zeta function ζ < > For the spectral zeta function ζ < > of H < >, we have the following: ζ (s) = j=1 p= (γ p λ j ) s 2 γ s 2 1 = ( + 1 γ s 2 1 γ ) (λ s j ) s 2 2 j=1 1 1 = ( 1 γ s 2 1 γ )ζ s S (s). 2 1 The geometric part 1 of the product structure is equal to zero. 1 γ 2 s 1 γ 2 s however, it represents an example of a hyperfunction known as the Dirac δ-hyperfunction. (Recall that γ > 1.) Michel L. Lapidus University of California, Riverside lapidus@math.ucr.edu Laplacians on Self-Similar Joint work Fractals withand Nishu Their LalSpectral University Zeta of Functions: California, Riverside Complex October Dynamics and Scripps 27, 2012 and College, Renormalizatio 27 Claremo / 41

28 Hyperfunctions The hyperfunctions are the distributional generalization of analytic functions. A hyperfunction f = [F +, F ] consists of two analytic functions, F + (z) and F (z), which are analytic in the upper and the lower half-planes respectively, such that the following limit makes sense (See, e.g., [Graf, 2010].) lim ɛ 0 [ F + (x + iɛ) F (x iɛ) ]. Michel L. Lapidus University of California, Riverside lapidus@math.ucr.edu Laplacians on Self-Similar Joint work Fractals withand Nishu Their LalSpectral University Zeta of Functions: California, Riverside Complex October Dynamics and Scripps 27, 2012 and College, Renormalizatio 28 Claremo / 41

29 The Dirac delta hyperfunction The Dirac delta hyperfunction on the real line R is defined by [ δ R (z) := 1 2πiz, 1 ]. 2πiz By definition, for x 0, δ(x) = lim ɛ 0 + = lim ɛ 0 + = lim ɛ 0 + ( ) F + (x + iɛ) F (x iɛ) ( ) 1 2πi(x + iɛ) 1 2πi(x iɛ) ɛ π(x 2 + ɛ 2 ) = 0. For x = 0, however, the above limit does not exist, and this is the point at which the delta function has an isolated singularity. Michel L. Lapidus University of California, Riverside lapidus@math.ucr.edu Laplacians on Self-Similar Joint work Fractals withand Nishu Their LalSpectral University Zeta of Functions: California, Riverside Complex October Dynamics and Scripps 27, 2012 and College, Renormalizatio 29 Claremo / 41

30 Factorization of ζ H< > In the case of the spectral zeta function ζ H< >, the geometric part of the spectral zeta function can be represented by the Dirac delta function with a suitable change of variable. Theorem (Lal Lap, 2012) The factorization of the spectral zeta function ζ H< > (s) of H < > is given by where w := γ s 2 the unit circle T. ζ H< > (s) = ζ S (s) δ T (w), (3) and δ T (w) = [δ + T (w), δ T (w)] is the Dirac delta hyperfunction on (Note that w < 1 for Re(s) > 0 and w > 1 for Re(s) < 0, since γ 4 implies that log γ > 0.) ichel L. Lapidus University of California, Riverside lapidus@math.ucr.edu Laplacians on Self-Similar Joint work Fractals withand Nishu Their LalSpectral University Zeta of Functions: California, Riverside Complex October Dynamics and Scripps 27, 2012 and College, Renormalizatio 30 Claremo / 41

31 A representation of the Riemann zeta function When α = 1 2, we obtain the usual Laplacian on the unit interval. Theorem (Lal Lap, 2012) When α = 1 2, the Riemann zeta function ζ is equal (up to a trivial factor) to the zeta function ζ ρ associated with the renormalization map ρ on P 2 (C). More specifically, we have ζ(s) = π s ζ ρ (s) = πs 1 2 s ζ S(s). (4) This is an extension to several complex variables of the result by A. Teplyaev (2007) which states that the Riemann zeta function can be written in terms of the zeta function of a quadratic polynomial of one complex variable. ichel L. Lapidus University of California, Riverside lapidus@math.ucr.edu Laplacians on Self-Similar Joint work Fractals withand Nishu Their LalSpectral University Zeta of Functions: California, Riverside Complex October Dynamics and Scripps 27, 2012 and College, Renormalizatio 31 Claremo / 41

32 Results for the Infinite Sierpinski Gasket We now extend the finite Sierpinski gasket to the infinite (or unbounded) Sierpinski gasket. Let k = {k n } n 1 be a fixed sequence, with k n {1, 2, 3} for all n 1. We construct a sequence SG (n) = Ψ 1 k,n (SG), where Ψ k,n = Ψ k1...k n := Ψ kn... Ψ k1. The infinite Sierpinski gasket is then defined by SG ( ) = SG (n), viewed as a blow-up of SG. The mth pre-gasket approximating SG (n) and SG ( ) are V m (n) = Ψ 1 k,n (V n+m) and V m ( ) = n=0 Ψ 1 k,n (V n+m), respectively. n=0 Michel L. Lapidus University of California, Riverside lapidus@math.ucr.edu Laplacians on Self-Similar Joint work Fractals withand Nishu Their LalSpectral University Zeta of Functions: California, Riverside Complex October Dynamics and Scripps 27, 2012 and College, Renormalizatio 32 Claremo / 41

33 Figure: An infinite Sierpinski gasket Michel L. Lapidus University of California, Riverside Laplacians on Self-Similar Joint work Fractals withand Nishu Their LalSpectral University Zeta of Functions: California, Riverside Complex October Dynamics and Scripps 27, 2012 and College, Renormalizatio 33 Claremo / 41

34 We next define the Laplacian (n) on SG (n) as follows: (n) u = f L 2 (SG (n), µ) iff E SG (n)(u, v) = (n) uvdµ, where E SG (n) SG (n) scaled copy of E n+m on V n+m for the finite Sierpinski pre-gasket. The pointwise Laplacian ( ) on SG ( ) can then be defined by the following pointwise limit: 5 n (n) u ( ) u as n. is a Michel L. Lapidus University of California, Riverside lapidus@math.ucr.edu Laplacians on Self-Similar Joint work Fractals withand Nishu Their LalSpectral University Zeta of Functions: California, Riverside Complex October Dynamics and Scripps 27, 2012 and College, Renormalizatio 34 Claremo / 41

35 Theorem (See, e.g., [Tep, 1998]) Let R(z) = z(5 4z). Then the spectrum of the self-adjoint operator ( ) acting on L 2 (SG ( ), µ) is pure point and the set of compactly supported eigenfunctions is complete. Furthermore, the set of eigenvalues is given by n= 5n R{Σ}, where Σ = { 3 2 } ( j=0 R j { 3 4 }) ( j=0 R j { 5 4 }) is the set of eigenvalues of the Laplacian µ on the finite SG, R(z) := lim m 5 m R m (z) and R m is the branch of the mth inverse iterate of R that passes through the origin. In particular, the spectrum of ( ) has the following form: n= j=0 5 n R(R j (z 0 )), (5) where z 0 = 3 4, 5 4. Every eigenvalue λ of ( ) can be expressed as λ = 5 n lim m 5 m R m (z m ) for some n Z, with z m in the spectrum σ( m ) of the finite mth Sierpinski pre-gasket. ichel L. Lapidus University of California, Riverside lapidus@math.ucr.edu Laplacians on Self-Similar Joint work Fractals withand Nishu Their LalSpectral University Zeta of Functions: California, Riverside Complex October Dynamics and Scripps 27, 2012 and College, Renormalizatio 35 Claremo / 41

36 Theorem (Lal Lap, 2012) The spectral zeta function ζ ( ) gasket SG ( ) is given by of the Laplacian ( ) on the infinite Sierpinski ζ ( )(s) = δ T (5 s 2 )ζ µ (s), (6) where δ T is the Dirac hyperfunction and ζ µ Laplacian on the finite SG. is the spectral zeta function of the ichel L. Lapidus University of California, Riverside lapidus@math.ucr.edu Laplacians on Self-Similar Joint work Fractals withand Nishu Their LalSpectral University Zeta of Functions: California, Riverside Complex October Dynamics and Scripps 27, 2012 and College, Renormalizatio 36 Claremo / 41

37 Thank you. Michel L. Lapidus University of California, Riverside Laplacians on Self-Similar Joint work Fractals withand Nishu Their LalSpectral University Zeta of Functions: California, Riverside Complex October Dynamics and Scripps 27, 2012 and College, Renormalizatio 37 Claremo / 41

38 Bibliography Fukushima, M., Shima, T., On a spectral analysis for the Sierpinski gasket, Potential Analysis 1 (1992), Graf, U., Introduction to Hyperfunctions and Their Integral Transforms: An applied and computational approach, Birkhäuser, Basel, Kigami, J., Analysis on Fractals, Cambridge Tracts in Mathematics, vol. 143, Cambridge University Press, Cambridge, Lal, N., Lapidus, M. L., Hyperfunctions and spectral zeta functions of Laplacians on self-similar fractals, J. Phys. A: Math. Theor. 45 (2012) , 14pp. (Also: e-print, arxiv: v2 [math-ph], 2012; IHES preprint, IHES/M/12/14, 2012.) Lal, N., Lapidus, M. L., The decimation method for Laplacians on fractals: spectra and complex dynamics, preprint, Michel L. Lapidus University of California, Riverside lapidus@math.ucr.edu Laplacians on Self-Similar Joint work Fractals withand Nishu Their LalSpectral University Zeta of Functions: California, Riverside Complex October Dynamics and Scripps 27, 2012 and College, Renormalizatio 38 Claremo / 41

39 Lapidus, M. L., Spectral and fractal geometry: From the Weyl Berry conjecture for the vibrations of fractals drums to the Riemann zeta function, in: Differential Equations and Mathematical Physics (C. Bennewitz, ed.), Proc. 4th UAB Internat. Conf. (Birmingham, March 1990), Academic Press, New York, 1992, pp Lapidus, M. L. and Maier, H., The Riemann hypothesis and inverse spectral problems for fractal strings, J. London Math. Soc. (2)52 (1995), (Also: C. R. Acad. Sci. Paris Sér. I Math. 313 (1991), ) Lapidus, M. L. and Pomerance, C., The Riemann zeta-function and the one-dimensional Weyl Berry conjecture for fractal drums, Proc. London Math. Soc. (3) 66 (1993), (Also: C. R. Acad. Sci. Paris Sér. I Math. 310 (1990), ) Lapidus, M. L. and van Frankenhuijsen, M., Fractal Geometry, Complex Dimension and Zeta Functions: Geometry and spectra of fractal strings, Springer Monographs in Mathematics, Springer, New York, (Second rev. and enl. ed., 2012.) Rammal, R., Spectrum of harmonic excitations on fractals, J. de Physique 45 (1984), Michel L. Lapidus University of California, Riverside lapidus@math.ucr.edu Laplacians on Self-Similar Joint work Fractals withand Nishu Their LalSpectral University Zeta of Functions: California, Riverside Complex October Dynamics and Scripps 27, 2012 and College, Renormalizatio 39 Claremo / 41

40 Rammal, R. and Toulouse, G., Random walks on fractal structures and percolation cluster, J. Physique Lettres 44 (1983), L13 L22. Sabot, C., Density of states of diffusions on self-similar sets and holomorphic dynamics in P k : the example of the interval [0, 1], C. R. Acad. Sci. Paris Sér. I: Math. 327 (1998), Sabot, C., Integrated density of states of self-similar Sturm Liouville operators and holomorphic dynamics in higher dimension, Ann. Inst. H. Poincaré Probab. Statist. 37 (2001), Sabot, C., Spectral properties of self-similar lattices and iteration of rational maps, Mémoires Soc. Math. France (New Series), No. 92, 2003, Sabot, C., Spectral analysis of a self-similar Sturm Liouville operator, Indiana Univ. Math. J. 54 (2005), Shima, T., On eigenvalue problems for Laplacians on p.c.f. self-similar sets, Japan J. Indust. Appl. Math. 13 (1996), Strichartz, R. S., Differential Equations on Fractals: A tutorial, Princeton University Press, Princeton, Michel L. Lapidus University of California, Riverside lapidus@math.ucr.edu Laplacians on Self-Similar Joint work Fractals withand Nishu Their LalSpectral University Zeta of Functions: California, Riverside Complex October Dynamics and Scripps 27, 2012 and College, Renormalizatio 40 Claremo / 41

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